The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter , or fixing the energy level instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function (where is a suitably rescaled version of ) with the event that there is no spectrum in an interval , in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.